Hypergeometric Function - 15.2 Definitions and Analytical Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.2.E1 F ⁡ ( a , b ; c ; z ) = ∑ s = 0 ∞ ( a ) s ⁢ ( b ) s ( c ) s ⁢ s ! ⁢ z s Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 Pochhammer 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}{\left(b\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}}}
\hyperF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\Pochhammersym{c}{s}s!}z^{s}		 

hypergeom([a, b], [c], z) = sum((pochhammer(a, s)*pochhammer(b, s))/(pochhammer(c, s)*factorial(s))*(z)^(s), s = 0..infinity)

Hypergeometric2F1[a, b, c, z] == Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Pochhammer[c, s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]
Failure Successful Skipped - Because timed out Successful [Tested: 300]
15.2.E2 𝐅 ⁡ ( a , b ; c ; z ) = ∑ s = 0 ∞ ( a ) s ⁢ ( b ) s Γ ⁡ ( c + s ) ⁢ s ! ⁢ z s scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}% \frac{{\left(a\right)_{s}}{\left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s}}}
\hyperOlverF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\EulerGamma@{c+s}s!}z^{s}		 | z | < 1 , ℜ ⁡ ( c + s ) > 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = sum((pochhammer(a, s)*pochhammer(b, s))/(GAMMA(c + s)*factorial(s))*(z)^(s), s = 0..infinity)

Hypergeometric2F1Regularized[a, b, c, z] == Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Gamma[c + s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]
Successful Successful -
Failed [25 / 216]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, 0.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[c, -2], Rule[z, 0.5]}

... skip entries to safe data
15.2.E3 𝐅 ⁡ ( a , b c ; x + i ⁢ 0 ) - 𝐅 ⁡ ( a , b c ; x - i ⁢ 0 ) = 2 ⁢ π ⁢ i Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ ( x - 1 ) c - a - b ⁢ 𝐅 ⁡ ( c - a , c - b c - a - b + 1 ; 1 - x ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑥 imaginary-unit 0 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑥 imaginary-unit 0 2 𝜋 imaginary-unit Euler-Gamma 𝑎 Euler-Gamma 𝑏 superscript 𝑥 1 𝑐 𝑎 𝑏 scaled-hypergeometric-bold-F 𝑐 𝑎 𝑐 𝑏 𝑐 𝑎 𝑏 1 1 𝑥 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-% \mathbf{F}\left({a,b\atop c};x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma% \left(a\right)\Gamma\left(b\right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c% -a-b+1};1-x\right)}}
\hyperOlverF@@{a}{b}{c}{x+\iunit 0}-\hyperOlverF@@{a}{b}{c}{x-\iunit 0} = \frac{2\pi\iunit}{\EulerGamma@{a}\EulerGamma@{b}}(x-1)^{c-a-b}\hyperOlverF@@{c-a}{c-b}{c-a-b+1}{1-x}		 x > 1 , ℜ ⁡ a > 0 , ℜ ⁡ b > 0 , | ( x + i ⁢ 0 ) | < 1 , | ( x - i ⁢ 0 ) | < 1 , | ( 1 - x ) | < 1 , ℜ ⁡ ( c + s ) > 0 , ℜ ⁡ ( ( c - a - b + 1 ) + s ) > 0 formulae-sequence 𝑥 1 formulae-sequence 𝑎 0 formulae-sequence 𝑏 0 formulae-sequence 𝑥 imaginary-unit 0 1 formulae-sequence 𝑥 imaginary-unit 0 1 formulae-sequence 1 𝑥 1 formulae-sequence 𝑐 𝑠 0 𝑐 𝑎 𝑏 1 𝑠 0 {\displaystyle{\displaystyle x>1,\Re a>0,\Re b>0,|(x+\mathrm{i}0)|<1,|(x-% \mathrm{i}0)|<1,|(1-x)|<1,\Re(c+s)>0,\Re((c-a-b+1)+s)>0}} 
hypergeom([a, b], [c], x + I*0)/GAMMA(c)- hypergeom([a, b], [c], x - I*0)/GAMMA(c) = (2*Pi*I)/(GAMMA(a)*GAMMA(b))*(x - 1)^(c - a - b)* hypergeom([c - a, c - b], [c - a - b + 1], 1 - x)/GAMMA(c - a - b + 1)

Hypergeometric2F1Regularized[a, b, c, x + I*0]- Hypergeometric2F1Regularized[a, b, c, x - I*0] == Divide[2*Pi*I,Gamma[a]*Gamma[b]]*(x - 1)^(c - a - b)* Hypergeometric2F1Regularized[c - a, c - b, c - a - b + 1, 1 - x]
Failure Failure Error Skip - No test values generated
15.2.E3_5 lim c → - n ⁡ F ⁡ ( a , b ; c ; z ) Γ ⁡ ( c ) = 𝐅 ⁡ ( a , b ; - n ; z ) subscript → 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 𝑐 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑛 𝑧 {\displaystyle{\displaystyle\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma% \left(c\right)}=\mathbf{F}\left(a,b;-n;z\right)}}
\lim_{c\to-n}\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{-n}{z}		 ℜ ⁡ c > 0 , | z | < 1 , ℜ ⁡ ( ( - n ) + s ) > 0 formulae-sequence 𝑐 0 formulae-sequence 𝑧 1 𝑛 𝑠 0 {\displaystyle{\displaystyle\Re c>0,|z|<1,\Re((-n)+s)>0}} 
limit((hypergeom([a, b], [c], z))/(GAMMA(c)), c = - n) = hypergeom([a, b], [- n], z)/GAMMA(- n)

Limit[Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]], c -> - n, GenerateConditions->None] == Hypergeometric2F1Regularized[a, b, - n, z]
Failure Successful Successful [Tested: 0]
Failed [25 / 36]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 3], Rule[z, 0.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[n, 3], Rule[z, 0.5]}

... skip entries to safe data
15.2.E3_5 𝐅 ⁡ ( a , b ; - n ; z ) = ( a ) n + 1 ⁢ ( b ) n + 1 ( n + 1 ) ! ⁢ z n + 1 ⁢ F ⁡ ( a + n + 1 , b + n + 1 ; n + 2 ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑛 𝑧 Pochhammer 𝑎 𝑛 1 Pochhammer 𝑏 𝑛 1 𝑛 1 superscript 𝑧 𝑛 1 Gauss-hypergeometric-F 𝑎 𝑛 1 𝑏 𝑛 1 𝑛 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;-n;z\right)=\frac{{\left(a% \right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1)!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z% \right)}}
\hyperOlverF@{a}{b}{-n}{z} = \frac{\Pochhammersym{a}{n+1}\Pochhammersym{b}{n+1}}{(n+1)!}z^{n+1}\hyperF@{a+n+1}{b+n+1}{n+2}{z}		 ℜ ⁡ c > 0 , | z | < 1 , ℜ ⁡ ( ( - n ) + s ) > 0 formulae-sequence 𝑐 0 formulae-sequence 𝑧 1 𝑛 𝑠 0 {\displaystyle{\displaystyle\Re c>0,|z|<1,\Re((-n)+s)>0}} 
hypergeom([a, b], [- n], z)/GAMMA(- n) = (pochhammer(a, n + 1)*pochhammer(b, n + 1))/(factorial(n + 1))*(z)^(n + 1)* hypergeom([a + n + 1, b + n + 1], [n + 2], z)

Hypergeometric2F1Regularized[a, b, - n, z] == Divide[Pochhammer[a, n + 1]*Pochhammer[b, n + 1],(n + 1)!]*(z)^(n + 1)* Hypergeometric2F1[a + n + 1, b + n + 1, n + 2, z]
Failure Failure
Failed [25 / 36]
Result: Float(undefined)+Float(undefined)*I
Test Values: {a = -3/2, b = -3/2, z = 1/2, n = 3}


Result: Float(undefined)+Float(undefined)*I
Test Values: {a = -3/2, b = 3/2, z = 1/2, n = 3}

... skip entries to safe data
 Successful [Tested: 180]
15.2.E4 F ⁡ ( - m , b ; c ; z ) = ∑ n = 0 m ( - m ) n ⁢ ( b ) n ( c ) n ⁢ n ! ⁢ z n Gauss-hypergeometric-F 𝑚 𝑏 𝑐 𝑧 superscript subscript 𝑛 0 𝑚 Pochhammer 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left% (-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}}}
\hyperF@{-m}{b}{c}{z} = \sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n}		 

hypergeom([- m, b], [c], z) = sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m)

Hypergeometric2F1[- m, b, c, z] == Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}, GenerateConditions->None]
Successful Successful - Successful [Tested: 300]
15.2.E4 ∑ n = 0 m ( - m ) n ⁢ ( b ) n ( c ) n ⁢ n ! ⁢ z n = ∑ n = 0 m ( - 1 ) n ⁢ ( m n ) ⁢ ( b ) n ( c ) n ⁢ z n superscript subscript 𝑛 0 𝑚 Pochhammer 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑛 superscript subscript 𝑛 0 𝑚 superscript 1 𝑛 binomial 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b% \right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}^{m}(-1)^{n}\genfrac{(}% {)}{0.0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c\right)_{n}}}z^{n}}}
\sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n} = \sum_{n=0}^{m}(-1)^{n}\binom{m}{n}\frac{\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}}z^{n}		 

sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m) = sum((- 1)^(n)*binomial(m,n)*(pochhammer(b, n))/(pochhammer(c, n))*(z)^(n), n = 0..m)

Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}, GenerateConditions->None] == Sum[(- 1)^(n)*Binomial[m,n]*Divide[Pochhammer[b, n],Pochhammer[c, n]]*(z)^(n), {n, 0, m}, GenerateConditions->None]
Successful Successful - Successful [Tested: 300]
15.2.E5 F ⁡ ( - m , b - m - ℓ ; z ) = lim c → - m - ℓ ⁡ ( lim a → - m ⁡ F ⁡ ( a , b c ; z ) ) Gauss-hypergeometric-F 𝑚 𝑏 𝑚 ℓ 𝑧 subscript → 𝑐 𝑚 ℓ subscript → 𝑎 𝑚 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left({-m,b\atop-m-\ell};z\right)=\lim_{c\to-m-% \ell}\left(\lim_{a\to-m}F\left({a,b\atop c};z\right)\right)}}
\hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{c\to-m-\ell}\left(\lim_{a\to-m}\hyperF@@{a}{b}{c}{z}\right)		 

hypergeom([- m, b], [- m - ell], z) = limit(limit(hypergeom([a, b], [c], z), a = - m), c = - m - ell)

Hypergeometric2F1[- m, b, - m - \[ScriptL], z] == Limit[Limit[Hypergeometric2F1[a, b, c, z], a -> - m, GenerateConditions->None], c -> - m - \[ScriptL], GenerateConditions->None]
Failure Successful Successful [Tested: 126] Successful [Tested: 126]
15.2.E6 F ⁡ ( - m , b - m - ℓ ; z ) = lim a → - m ⁡ F ⁡ ( a , b a - ℓ ; z ) Gauss-hypergeometric-F 𝑚 𝑏 𝑚 ℓ 𝑧 subscript → 𝑎 𝑚 Gauss-hypergeometric-F 𝑎 𝑏 𝑎 ℓ 𝑧 {\displaystyle{\displaystyle F\left({-m,b\atop-m-\ell};z\right)=\lim_{a\to-m}F% \left({a,b\atop a-\ell};z\right)}}
\hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{a\to-m}\hyperF@@{a}{b}{a-\ell}{z}		 

hypergeom([- m, b], [- m - ell], z) = limit(hypergeom([a, b], [a - ell], z), a = - m)

Hypergeometric2F1[- m, b, - m - \[ScriptL], z] == Limit[Hypergeometric2F1[a, b, a - \[ScriptL], z], a -> - m, GenerateConditions->None]
Failure Successful Successful [Tested: 0] Successful [Tested: 126]
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